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You've sort of included the solution to the paradox in your post. The paradox arose because it was insisted there is just one thing, infinity. Until it was pointed out that you can have different infinite things that definitely don't have the same number of things. The trick is that infinity isn't a number it's a limit. It's the limit beyond which you can't assign a value anymore. If you phrase it that way it makes sense you can have 2 sets which size are beyond counting but clearly have different sizes.

You've sort of included the solution to the paradox in your post. The paradox arose because it was insisted there is just one thing, infinity. Until it was pointed out that you can have different infinite things that definitely don't have the same number of things. The trick is that infinity isn't a number it's a limit. It's the limit beyond which you can't assign a value anymore. If you phrase it that way it makes sense you can have 2 sets which size are beyond counting but clearly have different sizes.

In the sense that a limit is a property and not a number exactly (though it can be one). So infinity isn't a number but a property of things. E.g. if you take a set of things the limit of it's size might be infinite. Rather than being an actual number you are counting towards. So things will tend towards infinity or have no upper limit, but that doesn't make infinity a number you can actually assign or manipulate like a normal number. Which means that 2 different sets of things tending to infinity can have different sizes.

It's the difference between saying "this list just keeps on going forever" and "the list has a size of X and X is the infinity number".

I get the idea that infinity isn't a number. I can also see how two infinite sets can contain different numbers of things. It's just the idea of infinity as a limit that doesn't limit anything, but maybe that's a semantic thing.

There's a concept in mathematics of a "limit" to a function. This is the value that the function approaches but never reaches as X approaches some value. The limit of f(x) can apply to any value of X, but one common value is infinity.

For example, suppose f(x) = 1/x. The limit of f(x) as x approaches infinity is 0, but it's never actually zero for any finite value, just a very small value. Conversely, as long as we're tossing around infinity and limits, f(0) = 1/0 is an undefined value, but the limit of f(x) as x approaches 0 is infinity.

Limits are highly useful things, since we get things like calculus from limits.

So yes, mathematical meaning is subtly different from the common one. I stopped learning maths before calculus came into the syllabus and haven't had a big enough incentive to learn it since. I am mildly curious about it though.

There's also some confusion because infinity doesn't actually exist. All things are countable, so an actual countable number can be assigned to any real problem or situation, even those with probabilities, which probably have the largest numbers we can conceive of. Infinity functions as an abstraction, an estimate. It indicates that I can replace the "infinite" quantity with any other arbitrarily large one. I often say things like "from the point of view of that (very fast) circuit, a second is effectively an infinite length of time." All this means is that it doesn't matter if we're talking about 1 second or 10 seconds or a year; the behavior of the circuit would be the same.

So yes, mathematical meaning is subtly different from the common one. I stopped learning maths before calculus came into the syllabus and haven't had a big enough incentive to learn it since. I am mildly curious about it though.

That's the nub of the thing. The paradox only exists the moment you try to do maths with the common concept of infinity. Then once you slot in the more technical difference the paradox is solved. It's weirdly true of a lot of things in life, where taking every day descriptions into maths/science/philosophy/logic they all break, or taking the "accurate" descriptions back into pub talk stops making any sense. An example of the reverse is eating peanuts in the pub and somebody pointing out they are in fact not nuts. Which is taking a specific botanical factual meaning of the word nut and applying it to a context where it makes no sense. In the context of a pub a peanut is a fucking nut, it's not somehow more intelligent to say otherwise.

Infinity might exist, space or time might be infinitive for example, but it mightn't.

That's kind of the point though. Any measurement we make or perception that we can actually have is bounded in time and space. Anything outside that is irrelevant. It might be that all the space just outside the visible universe is filled with packing peanuts.